Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2012-02-09
Physics
High Energy Physics
High Energy Physics - Theory
24 pages
Scientific paper
The correspondence between the semiclassical limit of the DOZZ quantum Liouville theory and the Nekrasov--Shatashvili limit of the N=2 (Omega-deformed) U(2) super-Yang-Mills theories is used to calculate the unknown accessory parameter of the Fuchsian uniformization of the 4-punctured sphere. The computation is based on the saddle point method. This allows to find an analytic expression for the N_f = 4, U(2) instanton twisted superpotential and, in turn, to sum up the 4-point classical block. According to the so-called Polyakov conjecture, the unknown accessory parameter is given by the derivative of the 4-point classical block with respect to the moduli of the 4-punctured sphere. It has been found that this accessory parameter is related to the sum of all rescaled column lengths of the so-called 'critical' Young diagram extremizing the instanton 'free energy'. It is shown that the sum over the 'critical' column lengths can be rewritten in terms of a contour integral in which the integrand is built out of certain special functions closely related to the ordinary Gamma function.
Ferrari Franco
Piatek Marcin
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